
theorem Th41:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema non void TA-structure for t being type of T for v
  being FinSequence of the adjectives of T st v is_applicable_to t for i1,i2
being Nat st 1 <= i1 & i1 <= i2 & i2 <= len v+1 for t1,t2 being type
  of T st t1 = apply(v,t).i1 & t2 = apply(v,t).i2 holds t2 <= t1
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TA-structure;
  let t be type of T;
  let v be FinSequence of the adjectives of T such that
A1: for i being Nat, a being adjective of T, s being type of
  T st i in dom v & a = v.i & s = apply(v,t).i holds a is_applicable_to s;
  let i1,i2 be Nat such that
A2: 1 <= i1 and
A3: i1 <= i2 and
A4: i2 <= len v+1;
  consider j being Nat such that
A5: i2 = i1+j by A3,NAT_1:10;
  let s1,s2 be type of T;
  defpred P[Nat] means i1+$1 <= len apply(v,t) implies for s being
  type of T st s = apply(v,t).(i1+$1) holds s <= s1;
A6: len apply(v,t) = len v+1 by Def19;
A7: for i being Nat st P[i] holds P[i+1]
  proof
    let i be Nat such that
A8: P[i] and
A9: i1+(i+1) <= len apply(v,t);
    i1 <= i1+i by NAT_1:11;
    then
A10: 1 <= i1+i by A2,XXREAL_0:2;
A11: i1+(i+1) = i1+i+1;
    then i1+i <= len v by A6,A9,XREAL_1:6;
    then
A12: i1+i in dom v by A10,FINSEQ_3:25;
    then v.(i1+i) in rng v by FUNCT_1:3;
    then reconsider a = v.(i1+i) as adjective of T;
    i1+i < len v+1 by A6,A9,A11,NAT_1:13;
    then i1+i in dom apply(v,t) by A6,A10,FINSEQ_3:25;
    then apply(v,t).(i1+i) in rng apply(v,t) by FUNCT_1:3;
    then reconsider s = apply(v,t).(i1+i) as type of T;
A13: apply(v,t).(i1+i+1) = a ast s by A12,Def19;
A14: a ast s <= s by A1,A12,Th20;
    s <= s1 by A8,A9,A11,NAT_1:13;
    hence thesis by A13,A14,YELLOW_0:def 2;
  end;
  assume that
A15: s1 = apply(v,t).i1 and
A16: s2 = apply(v,t).i2;
A17: P[0] by A15;
  for i being Nat holds P[i] from NAT_1:sch 2(A17,A7);
  hence thesis by A4,A5,A6,A16;
end;
